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arXiv · 1107.3933

Rigidity percolation on the square lattice

Wouter G. Ellenbroek·Xiaoming Mao
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Abstract

The square lattice with central forces between nearest neighbors is isostatic with a subextensive number of floppy modes. It can be made rigid by the random addition of next-nearest neighbor bonds. This constitutes a rigidity percolation transition which we study analytically by mapping it to a connectivity problem of two-colored random graphs. We derive an exact recurrence equation for the probability of having a rigid percolating cluster and solve it in the infinite volume limit. From this solution we obtain the rigidity threshold as a function of system size, and find that, in the thermodynamic limit, there is a mixed first-order-second-order rigidity percolation transition at the isostatic point.

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Wouter G. Ellenbroek, Xiaoming Mao. 2011-07-20. Rigidity percolation on the square lattice. https://doi.org/10.1209/0295-5075%2F96%2F54002

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